Density functional theory and optimal transportation with Coulomb cost

Density functional theory is a widely used variational model of the electronic structure of molecules, clusters, and bulk matter (it underlies most current numerical electronic structure predictions in physics, chemistry, materials science, molecular biology). There exists an 'exact' density functional whose minimizer co-incides with the correct electron density predicted by full many-body quantum mechanics (the so-called Hohenberg-Kohn functional). But its construction is too indirect to be computationally usable. Explicit functionals are obtained via 'closure relations' expressing the electron pair density or the N-point density in terms of its one-point marginal (alias single-particle density).

In the semi-classical limit, there emerges an exact closure relation: the N-point density is the solution of an N-body optimal transport problem with Coulomb cost. An interesting and unusual feature of this problem is that the cost decreases (rather than increases) with distance. The limit problem is known in the physics literature since 1999, but the fact that it has the structure of an OT problem and can be usefully analyzed via OT theory was first noticed by us in 2011 (C.Cotar, G.F., C.Klueppelberg, arxiv.org/abs/1104.0603, 2011 and CPAM 66, 548-599, 2013).

In the talk I will begin by informally explaining the connection quantum mechanics -- density functional theory -- optimal transport (not assuming expertise in any of these fields). I will then discuss

• qualitative theory of OT with Coulomb cost, including the question whether ''Kantorovich minimizers'' must be ''Monge minimizers'' (yes for 2 particles, open for N particles)
• exactly soluble examples, including N particles, 2 sites (arxiv.org/abs/1304.0679, 2013, with C.B.Mendl, B.Pass, C.Cotar, C.Klueppelberg)
• the limit of infinitely many particles (Preprint, 2013, with C.Cotar and B.Pass).